Theoretical Computer Science 65 (19X9) 197-212

North-Holland

197

STATISTIC AND ERGODIC PROPERTIES OF MINKOWSKI’S DIAGONAL CONTINUED FRACTION

Cot- KRAAIKAMP

Insrirute of Mathemalics, Amsterdam Univer.Gty, Roetersstraat 15, 1018 WI3 Amsterdam,

The Netherlands, and U.F.R. de MathPmatiquer, Universith de Provence, 3 Place Victor Hugo,

13331 Marseille, France

Received March 1988

Revised June 1988

Abstract. Recently the author introduced a new class of continued fraction expansions, the

S-expansions. Here it is shown that Minkowski’s diagonal continued fraction (DCF) is an

S-expansion. Due to this, statistic and ergodic properties of the DCF can he given

Contents

1. lntroduction..................................................................

2. S-expansions . .._............._....._......_......_._....._._..._._...,,......

3. Minkowski’s diagonal expansion as an S-expansion..

4. The distribution of some sequences connected with Minkowski’s diagonal expansion.

5. The DCF-algorithm revisited. References _._......_...._....._.._.,_._....._._..,_._._,_,,,..__._,.._._,.._.

197

200

203

20.5

209

211

1. Introduction

Let x be an irrational number between 0 and 1. let

(1.1) 1

b,+ 1

b2+ 1

. .

be its expansion as a regular continued fraction, denoted by RCF, and let (p,,/ qn)z=-,

be the corresponding sequence of convergents. Here q_, = 0.

1.2. Definitions. The operator T:[O, l]+ [0, l] is defined by

77x:=x-‘-[X ‘1, XfO,

TO := 0.

0304-3975/X9/$3.50 (Q 1989, Elsevier Science Publishers B.V. (North-Holland)

198 C. Kraaikamp

Put

T,, := T”(x): the nth-iterate of T on x, where n 2 1 and T,,:= x

and

v,, := 4,,~,/$?> n 2 0.

Notice that (T,,, V,,), -cl is a sequence in 0, where 0 := [0, l] x [0, 11. Let the sequence

of regular approximation constants 0, = t?,,(x), n 2 -1, be given by

8,, := 9,,lqnX-P,11, n 3 -1.

Then we have

O<fI,,<l, nzo,

and

(1.3) o,, = A> n SO; ,I ,1

see e.g. [7, p. 29, Eq. (1 l)].

Furthermore we have the following classical theorems of Legendre and Vahlen.

1.4. Theorem (Legendre). Let x be an irrational number and let P, Q E Z such that

(P, Q) = 1, Q > 0 and 0 = Q1 Qx - PI < 4. Then P/Q is a regular convergent ef x.

1.5. Theorem (Vahlen). For all kEN and X@CD we have min(&, &+,)<i.

1.6. Definitions. Here and in the following, [a,,; ~,a,, FLU?, . . .] is the abbreviation

of

a0 + &I

aI + &2

a,+ F3

a,+ . . .

where a,, E E, a, E N, i 3 1 and F, E {*l}, i E N. We call [a,,; F, a,, . . .] a semi-regular

continued fraction expansion (SRCF) in case F, + a, 2 1, F,, , + a, 2 1 for i b 1 and

F,~+, + a, 2 2 infinitely often.

Let x be an irrational number. Consider the sequence u of all irreducible rational

fractions P/Q, with Q > 0, satisfying

P

I I

1 1 x-- <__ Q 2Q2’

ordered in such a way that the denominators form an increasing sequence.

From Legendre’s Theorem 1.4 it follows that u consists exactly of those regular

convergents pl/qr for which fIA < :. Due to this and Vahlen’s Theorem 1.5 we see

that u is an infinite subsequence of the sequence of regular convergents of x. In

[13, Section 411 it is shown that there exists a unique SRCF-expansion of x such

that u is the sequence of convergents of this expansion of x. By definition, this is

Minkowski’s diagonal continued fraction expansion (DCF) of x; see also [II)].

Statistic and ergodic properties o.f Minkowski’s DCF 199

1.7. Definition. Let the irrational number x have the continued fraction expansion

[a,; clalr. .] and suppose that for a certain k 2 0 one has ak+, = 1 and F~+, = &A+? = 1.

The operation by which this continued fraction is replaced by

[a,; clal, czar,. . , &kc4 + 11, -(&+I+ I), F!%+34+3,. . .I,

which is again a continued fraction expansion of x, is called the singulurization of

the partial quotient a,,, equal to 1, see also [3, Section 11.

1.8. Remarks. (i) The singularization of a partial quotient equal to 1 is based upon

the equality

1 A+ =A+l- ’

1 1+-

B+l+t’

B+5

where B, 5 > 0.

(ii) Notice that if we repeat this singularization operation, we can never singular-

ize two consecutive partial quotients.

(iii) Let (A,/&),_,_, be the sequence of convergents of expansion (1.6) and

( Cn/Dn)n_Z_l that of the expansion obtained by singularizing in (1.6) an ok+, equal

to 1. Then the sequence (C,/D,),,_, is obtained from the sequence (A,/B,),x_,

by skipping the term Ak/ Bk.

1.9. Theorem. Minkowski’s DCF-expansion qf an irrational number x is obtained

.from the RCF-expansion of x by singularizing all those regular partial quotients b,,,

for which 13~ > :.

Proof. The theorem is an immediate consequence of the observation that ok > $

implies b,,, = 1 and of Vahlen’s Theorem 1.5. 0

1.10. Definition. Let x be an irrational number and let [a,,; ~,a,, ~~a~,. . .] be its

DCF-expansion. We denote the sequence of DCF-convergents of x by

m(x) ~ na-1 s,,(x)’

or shortly by :, nz-1.

The DCF-approximation constants 0, = O,(x), n 3 -1, are defined by

0, := s,Ir,x-ss,I, n>-1.

In this paper we show that the one-sided shift-operator connected with the

DCF-expansion comes from a certain dynamical system. Due to this, the distribution

of some sequences connected with the DCF can be given. In fact, the DCF is an

example of a wider class of continued fraction expansions, so called S-expansions;

see also [9]. For S-expansions the underlying dynamical system can be given; a

short description of these expansions will be given in Section 2. Finally, it is shown

that the DCF-expansion of a quadratic surd is periodic.

We conclude this section with an example.

200 c‘. Kruaikamp

1.11. Example. Let x = (-39 +-)/X3 = 0.2764. . . One has RCF(x) =

[O; 3, 1, 1, 1, 1, 1, 11. Hence

n -1 0 1 2 3 4 5 6 7 8 9 . . .

h,, - 03 1 1 1 1 1 1 3 1 . Prz 1 0 1 1 2 3 5 8 13 47 60 .

9,1 0 1 3 4 7 11 18 29 47 170 217 .

o,, 0 x 0.51.. 0.42.. 0.45.. 0.45.. 0.41.. 0.52.. 0.23.. 0.52.. 0.41.. . . .

Thus,

n -101 2 3 4 5 6 7 .

F,, _ - 1 -1 1 1 1 -1 -1 .

a,, _ 04 2 1 1 2 5 2 . r,, 1 01 2 3 5 13 60 107 .

s,, 0 1 4 7 11 18 47 217 387 . or, 0 x 0.42.. 0.45.. 0.45.. 0.41.. 0.23.. 0.41.. 0.45.. .

. .

.

. .

One finds that DCF(x) = [O; 4, -2, 1, 1,2, -51. The periodicity of this expansion

does not follow from the above short calculation but from the algorithm given in

Section 5.

2. S-expansions

Fundamental in

[ll, 121.

the theory of S-expansions is the following theorem; see also

2.1. Theorem. Let

mea.sure on (0, B)

.T( x, .v) :=

B he the collection qf Borel-subsets sf R and p the probabilit)

with density (log 2) ‘( 1 + .uJ’) ‘. Dejine the operator .“?I R + R b?

(Tq(_Y+[x~‘])~‘), (X,?‘)ER.

Then (0, B, p, 9) forms an ergodic s~~stem.

A simple way to derive a strategy for singularization is given by a singularization

area S.

2.2. Definition. A subset S from R is called a singularizarion area when it satisfies

(i) SE B is /I-continuous;

(ii) sc[i, l]X[O, 11;

(iii) ( .TS) n S = (4.

2.3. Theorem. Let S be a singularization urea. Then

log 2g log G O~~(S)~~== 1 -p= 0.30575 . . .

log 2 log 2

201

For a proof of this, see [9].

Here and in the sequel we put

g:=fi-l G.=A+l -=g+1=g-‘. 2’ .2

2.4. Definition. Let S be a singularization area and x an irrational number. The

S-expansion of x is obtained from the regular expansion of x by singularizing b,,,,

if and only if (T,,, V,,) E S. Here T,, and V,, are defined as in Definition 1.2.

2.5. Definition. Let S be a singularization area, x an irrational number and let

[a,,; ~,a,. e2a2,. . . ] be the S-expansion of x. The shift t which acts on X-Q,, is

defined by

t([O; &,a,) Ez(1?, . . .]) :== [O; eza2, FJU3,. . .].

Moreover, let I, be the kth iterate of t on x-u,, and let vA := s~-,/Q, k 20, where

Q is the denominator of the kth S-convergent of x.

2.6. Remark. By definition, t,, = [O; E~+,uA+, , cl, 2uk+lr.. .]. One easily sees that

vI = [O; a,, ehuL_, , . . , &?a,] and that the numerators and denominators of the

sequence ( Y~/s~)~ _, , the sequence of S-convergents, satisfy the following recurrence

relations:

r_, := 1, yo:= 00, r, := a,,r,,__, + e,,r,,_, _, n31,

SK, := 0, so:= 1, s, := a,,s,,_, + F,,S,,_7 _I n 3 1.

2.7. Lemma. With S and x as in De$nition 2.5, ( p,l/q,,)n __, the sequence of regular

convergents and (T~/Q)~ ._, the sequence of S-convergents qf x, we have

in- ,(bn + L)+P,-, x=

p,z + T,p,,- ,

qn~,(b,,+T,,)+qn-,=q,,+T,,q,l~,’ nzl

and

From Remark 1.8(iii), Definition 2.4 and Lemma 2.7 one easily derives the

following theorem.

2.8. Theorem. Using the same notation as in Dqjinition 2.5 and putting A := f2\S,

A-:= 9s and A’:= A\A-, we have

(i) (T,,, V,,)E S * p,,/q,, is not an S-convergent;

(ii) p,,/q,, is not an S-convergent =3 both p,l_,/q,,_, and p,,, ,/q,,+, are

S-convergents;

202 C. Kraaikamp

(iii) (T,, V,,)E A’ G 3k: rk- I =nml, rh =pn

and

t,, = T,, (hence eLiI := sgn(t,) = +l)

VI, = v,, ;

tk=-T,,/(i+T,,) (hence&h+l=-l)

UI, = 1 - v,,.

2.9. Remarks. Define the transformation 9’: A + A by

where A is defined as in Theorem 2.8. Due to the fact that % is an induced

transformation, we now have that (A, B, p, Y’) forms an ergodic system. Here p is

the probability measure on (A, B) with density

1 1

,u(A) log 2 (lt tv)”

see e.g. [ 141. Since h(T), the entropy of the RCF, equals

h(T) =z 6log2’

see [ll], we have, due to a formula of Abramov, h(Y) = h( T)/p(A), see [l]. It is

now natural to consider the following definition.

2.10. Definition. Let the map M : A + [w’ be defined by

M(T, V):= CT, VI, CT, VIEA+, (-T/(l+T),l-V), (T, V)EA-.

2.11. Theorem. Let S be a singularization area and put Rs := M(A) = At u M(A-).

Let again B be the collection of Borel-subsets of R, and let p he the probability measure

on (as, B) with density (p(A) log 2)~‘(1+ tu)-‘. Define the map T: R,+fl, by

T( t, v) := M( .Y( M -‘( t, u))). Then 7 is conjugate to Y by M and we have

(i) ( tk, q) E fi?5, tlk 3 0;

(ii) (O,Y, B, p, T) forms an ergodic system;

(iii) ?I(T) = h(Y).

Moreover we have the following theorem.

Stafistic and ergodic properties of Minkowski’s DCF 203

2.12. Theorem. Let the map f: 0, + R be dejined by

f(4 v):=ItP’I-?(f, v), (4 v)gfls,

where r, is theJirst coordinate function of r. Let b(t) := [t-l], Vt E R, t # 0. Using the

same notations as in Theorem 2.11 we now have

(i) f(t, v)= b(t)+1’

1

b(t), when sgn( t) = 1, 9( t, v) & S,

when sgn( t) = 1, .T( t, v) E S,

b(-r(l+t))+l, when sgn(t)=-1, F(M-‘(t,v))&S,

b(-t/(l+t))+2, when sgn(t)=-1, Y(M-‘(t,v))ES;

(ii) 44 0) = (It-‘1 -f(4 v), (sgn(t)v+f(t, v))Y’), V(t, V)E QT.

A consequence of this is the following corollary.

2.13. Corollary

(i) f(4 v) E N, v(t, 0) E f?~.

(ii) aA+, =f( tk, vk), VkzO, where (to, v,)=(x-a,,O).

For proofs and more results on S-expansions, see [9].

3. Minkowski’s diagonal expansion as an S-expansion

From the definition of Minkowski’s diagonal continued fraction (DCF)

formula (1.3) it follows at once that the DCF is an S-expansion with

s=s”cF:= (T, v,d&>; { I (see Fig. 1). Notice that we now have

and

T 1 V (T, V)E@-<- -<I T>O, V>O

l+TV 2’1+TV 2’

Fig. 1

204 CI Kranikamp

and

(l+T)(l-V)<i _l<T<O V>O l+TV 2’2 ’

(see also Fig. 1) where A’, K are defined as in Theorem 2.8 and M is defined as

in Definition 2.10. Since PL(S,)~.~ ) = 1 -l/(2 log 2) (see [4, p. 2861) we find the

following theorem (see also Definition 2.5).

3.1. Theorem. The two-dimen.sional ergodic system ,for the DCF is (R,s,,c, , B, p, r)

where p is the probability measure on O,s,,c, with densit~~ 2/( 1 + tc)‘.

In some cases, e.g. Nakada’s a-expansions, Bosma’s OCF, which are all examples

of S-expansions, it is possible to obtain an explicit expression for .f( t, v). See also

[9,5]. In these cases, one no longer depends on the RCF to obtain the S-expansion

of X. Since S,,,-, has relatively smooth boundaries, it is possible to obtain an explicit

expression for ,f’=,f;,c.b, using Remark 2.6 and Lemma 2.7. Indeed we have the

following theorem.

3.2. Theorem. For all (t, U) E 05,,c, ,

,f(t,v)= It ‘If [

[It ‘l]+sgn(t) u- 1 1 2([lt ‘l]+sgn(t). 0)-l

Proof. Let n EN. Put

<,<~,.?(f,U)CS,,<., n

<t&.Y(t,U)ELs,)(p n

and

(t, u)E M(_l-);&<- -’ <’ Y(M-‘(t, v)).@S,,~.~ l+t n”

We will only prove the theorem for (t, u) E B := BT; the other cases are proved in

the same way.

A simple calculation yields, using Theorem 3.1 and the definition of .Y,

Statistic and ergodic properties of Minkowski’s DCF 205

From Theorem 2.13 we have f( t, v) = 1; hence we must show that

[

lt_,,+ [Ifp'll+w(f)~ u-1 2([(t-‘l]+sgn(t). 0)--l 1

= 1.

Now

[

,t-,,+ [Ifm’ll+w(r). v-1 2([)t-‘I]+sgn(t) * u)-I] =[++&l]

and we have, due to (t, v) E B,

1+2u 1 -----<t<- 3v+2 2-v’

Since

2-v+- u >l for(t,v)EB, 2u+1

we thus find

[;+&]=I. 0

3.4. Remark. Notice that we also have that II(T) = $T*.

4. The distribution of some sequences connected with Minkowski’s diagonal expansion

Only a few metrical results are known for the DCF; they are to be found in [4].

These results are

(i) for almost all x the sequence (8,(x)), .(, is uniformly distributed over the

interval [0, $1;

(ii) let x be an irrational number and let the monotonic function k : N+ N be

such that

rn PA(~) _=-

SF, G(n)’ n=l,2,...;

then one has, for almost all x,

lim k(n) -=2log2=1.3862....

n-,x n

Using the theory of S-expansions we are able to extend this considerably. For

instance, we will obtain for almost all x the distribution of the sequences

(Ok_, , OA)k -, , (Ok-, + Ok)I, ., and the relative frequency of the partial quotient 1.

Let [a,,; &,a,, E2a ?, . . .] be the DCF-expansion of the irrational number x and

(rL/~l)k-E, be its sequence of DCF-convergents. From Theorem 3.1 we derive, using

techniques analogous to those used in [6, 81, the following theorem.

206 C. Kraaikamp

4.1. Theorem. For almost aff irrational numbers

( tL, v~)~ _(, is distributed over Rsl,l, =: fls according

h(x, y) = 2/( 1 +x-y)‘.

Since

x, the two-dimensional sequence

to the density,function h, given by

Ihl Ok = O,(x):= s,)s,x-r,l=- l$_fkVl,

VkzO,

it is natural to consider the map 4 : R, + R’, defined by

$b(t,v):= ( u - Itl > l+lv’l+fv ’

V(t, V)EO,

Let &, := $({(t, v) E 0,; sgn( t) = +I}), d2:= (cr({( t, v) E fl,s; sgn( t) = -1)). A simple

calculation shows that

.d, = {(x, y) E CR”; 0 =z x, ?’ G $},

,FPZ={(X,y)E[W2;O~.X~~, Oay~~,O~(x-y)‘+(x+y)~~}

(see also Fig. 2).

Moreover, the absolute value of the Jacobian J of (CI on 0 equals

1 - tv

(1 + tv)’

Fig. 2.

Stafistic and ergodic properties of Minkowski’s DCF

and

IJI_’ 2 = 2

(l+ru)* Jl-4tlJ/(1+tu)”

Hence we have proved the following theorem.

4.2. Theorem. For all irrational numbers x the sequence (Ok-, , Ol)k_,, is a sequence

in ~4 := -cP, CJ d2 and for almost all x this sequence is distributed over ~4 according to

the density .function d, where d := d, + d2 and

2

d,(x, y):=

J_’ (X,Y)E~~17

0 > (x,y)~f,;

2

%qFGjq’ (X,.Y)E&,

dz(x, y) := 0 3 (X,Y)Eh.

Using techniques analogous to the ones used in [8, Section 41, we find the following

corollary.

4.3. Corollary. For almost all x the sequence (Ok_, + Ok)kXO is distributed over the

interval [0, l] according to the density function m, where m := m, + m, and

I 1+a

log - 1-U’

O<a<S,

m,(a) = 2 ,ogfi+jTi-a)

Jm , i<a<l;

(2 arctg a, O<a<$,

m’(a)X{2arctgJm, :<a<:.

For a picture of m, and mz, see Fig. 3.

In a similar way one could determine the distribution of the sequence

(Ok_, - Ok)L _,, over the interval [-i, $1.

A classical result (see [2]) states that for the RCF one has, for almost all x,

lim NP’#{j< N; b, = 1) = 1

~ log ; = 0.41503 . . . . N-u’ log 2

In Example 1.11 we saw that not every regular partial quotient b,,, equal to 1

disappears in the diagonal expansion of x, as is e.g. the case in the nearest integer

continued fraction expansion of x. One may ask how many partial quotients equal

0.0 0.2 o.* 0.6 1.0

Fig. 3.

to 1 “survive” in the DCF. Note that a regular partial quotient b,,, equal to 1 does

not disappear in the diagonal expansion of x if and only if (T,, V,)r Ai and

Y( T,, V,) ~6 S,,,-,. Hence we find the following theorem.

4.4. Theorem. A regular partial quotient hk,, equal to 1 does not disappear in the

diagonal expansion qf’x fund only if (T,, V,) E B, with B as in (3.3).

For a picture of B, see Fig. 1.

Note that the hyperbolae

(l-t)(l+u) 1 and t 1

1+ IV 2 1+tl_-2

are tangent to each other in (4, 0) and that

1 P(B) =p

log 2

Thus we see that in the singularization process leading to the DCF, only 4.7% of

the original 41% of partial quotients equal to 1 is saved (for almost all x). After a

normalization we therefore find this result.

Statistic and ergo&c properties of‘ Minkowski’s DCF 209

4.5. Theorem. Let x be an irrational number and let (ak)k,,, be the sequence of

DCF-partial quotients of x. Then,

lim N~‘#{j~N;a,=1}=2(log(fi-l)+fi-4)=0.0656... a. e. N-I-

5. The DCF-algorithm revisited

In this section we study the values of 0, corresponding to a block of m consecutive

partial quotients equal to 1 in the regular expansion of an irrational number x. The

result, a simplification of a method described in [13, p. 183-1841, enables us to

obtain the diagonal expansion of x from the sequence (b,,),, _,, of regular partial

quotients of x. We show that the DCF-expansion of a quadratic surd is periodical.

Before stating the main result Theorem 5.3 of this section, we mention two useful

tools. A simple consequence of Definition 1.2 of the RCF-operator T is the following

lemma.

5.1. Lemma. Let RCF(x) = [b,,; b,, . , bk,. . .], RCF(x’) = [bo; b,, . . . , b;, . .],

where b, # b;. _ lf k is even, then

b,<b; Q x<x’;

_ if k is odd, then

b,<bj, w x)x’.

From Definition 1.7 and Remark 1.8(i) the next lemma follows at once.

5.2. Lemma. LettE(O,l) and RCF([)=[O; L?,,Bz ,... 1, whereB,#l. Then l-l=

[O; 1, B, - 1, B,, . . .].

In the following we denote a block of m consecutive l’s by 1”‘.

5.3. Theorem. Let O<x<l be an irrational number, RCF(x) =

[O; b,, . . , b,,, l”‘, b,+,,,+, , . . .], where b,, b,,,,,, # 1 .for n 2 1, and b,,, # 1 .for n = 0.

- Ifm=l orn=O, then

o,, > 1.

_ [fm>l andnzl, then

0, >: if and only jf [O; l”‘-‘, b,+,,,+,, . . .] < [O; 1, b,, - 1,. . , b,],

0 ,,++, >$ if and only if [O; l”‘-‘, b,, . . . , b,] < [O; 1, b,+,n+, - 1,. . .], and

0 n+P <$ ,for O<e<m-1.

210 C. Kraaikamp

Proof. From the definition of .Y one easily derives the following (see also

Theorem 2.1).

l In case m=l, (T,,, V,)r[3,l]\qx[O,I]cS,,~,. Hence,

(5.4) 8,) > :.

l In case m>l, we find for O<e<m-1, (T,,,, V,, , .) E [i, <]\O x [i, 11. Since

[A, :]\Q x [A, 11 n S,)(.r = M, we find

(5.5) 0,<; forO<e<m-1.

In general we have, for 0~ e G m - 1, due to Definition 1.2 and formula (1.3),

(5.6) fL+<, =(~+v’)~‘=(l+~,+~,+‘+v.“)~~

= (l+ [O; l”‘P’m’, b,,,,,,,, , . . .] + [O; l’, b,,, . ) b,])- ‘.

Notice that (5.4) and (5.5) are immediate consequences of (5.6). We now have, due

to Lemma 5.2, eq. (5.6) and b,,, b,, t,,l+I f 1,

and H,,>i iff [O; 1”’ ‘, b,,+,“,+,,. . .I<[% 1, h-1, km,,. . , b,l,

0 ,I + ,,I I >l iff [O; l”‘-‘, b,,, . . , hl< LO; 1, b,,+mt, - 1, b,,+t,,+z,. . .I,

which proves the theorem. II

From Theorem 1.9 and Theorem 5.3 we have a corollary.

5.7. Corollary. Minkowski’s DCF-expansion of an irrational number O< x < 1 is

obtained,from RCF(x) by singularizing each regular partial quotient bl+, equal to 1

which satisfies one qf the ,four ,following conditions:

’ bk+,= b,; that is, the case k = 0,

l b,, bA,?# 1, k>O,

l bl # 1, b,+,= 1 and [O; blt3,. . .]>[O; b,,-1, bk_ ,,..., b,], k>O,

l b, = 1, bL,r # 1 and [O; b,, , , . , b,] > [O; blt2- 1, b,,,, . .], k > 0.

5.8. Remark. Since the conditions

LO; b k+?, . .] > [O; b, - 1,. . . , b,] and

[O;b I,..., b,l>[O;b~+2~l,b,,,3,...1

are easily checked, due to Lemma 5.1, it is relatively easy to obtain the diagonal

expansion of x from its sequence of regular convergents.

Statistic and ergodic properties c$Minkowski’s DCF 211

Let 0 < x < 1 be a quadratic irrational number. Then, by a classical theorem of

Lagrange, the regular expansion of x is ultimately periodic, i.e.

RCF(x) = [O; b,, . . , b,,,j, b,,+, , . . , b,,+J. We take n, and the period length La 1 both minimal. Put bj,r = bn,,+,+kL, where

k EN, j 2 0. Suppose that for some i E { 1,. . , L} we have: b,,,),, = 1. Now if we want

to obtain the DCF(x) from the RCF(x) we have to distinguish the following cases,

due to Corollary 5.7.

(i) If b,-,,k, b,+, k 1 # 1, then b,,A must be singularized, for all k EN.

(ii) If b,_,,k = b,+),k = 1, then b,,L must not be singularized, for all k E N.

(iii) If b,_,,k # 1, b,,,,, = 1, then b,,A must be singularized if and only if

(5.9) IO; Cl, c2 ,... I>[@ d, ,..., 4(,+,+~rl

where

[O; c , , . .I = [O; h,+,+z, . . . , b,,,,+L, h,j+, , . . . , &<,+,+,I and

[O; d,, . . . , dq,+,+ul= LO; k,,,, - 1,. . . , b, > . . , hl.

In particular we have, in case L = 3, c, = d, + 1 and in case L > 3,

c, = b q+i+2 > CL-~= b,,+,~,, d, = b,,+, -1 and dLm2= bn,,+1+2.

Therefore it follows from Lemma 5.1 that (5.9) is equivalent to

(5.10) [O; &,+i+z, . . , b,,,tL, b,,,+, , . . . , b,,,,+,~,l

> LO; b,~, tp, - 1, . . . , h,,+L, kc,+, , . . , b,~,,+i+Jr

which is independent of k.

Thus we see that b,,,, must be singularized, for all k E N, if and only if (5.10) holds.

(iv) If b,-,,,, = 1, b,+,,k # 1, then we find, in the same way as in (iii), that b,,,, must

be singularized for all k E N if and only if it holds that

[O; b,,,,+,m2, . . . , b,,u+,, b,,,j+L, . . . , &+,+,I

> LO; bn,,+i+l - 1, bn,,+t+2, . . . 3 b,tt,+,mll.

A direct consequence of (i)-(iv) and Corollary 5.7 is the following theorem.

5.11 Theorem. Tile Minkowski DCF-expansion of a quadratic irrational number is

periodic.

5.12. Remark. A different proof of this theorem can be found in [13, Section 411.

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